With suburban sprawl, it's likely the only places within walking distance (i.e. "kid-reachable") are other houses, maybe a park. If managed well, a dense(r) city can have a forest or other semi-natural area nearby to everyone.
I have acres of woods _in my backyard_. There are also no homeless people sleeping on the benches.
I live in Prague, my parents in a nearby town. Walking 10 minutes gets me either to a metro station or an "urban forest"/heavily wooded park called Stromovka. I think it's a very appropriate place for kids to play in nature. My parents have walking/biking access to similar places.
Not every place in Prague is this close to nature, but it's common enough that someone seeking that kind of housing can get it at basically no premium.
Mah suburbia people do not even grasp the alternatives they are missing
Offhand, from relatively recent experience:
Singapore and Hong Kong contain forests, and when living there I regularly walked to them. Hong Kong is more than half green space.
The east side of Cologne is around an hour's walk from a proper forest.
Central Salzburg is an easy walk from several forests.
Hiroshima is surrounded by gorgeous forests, easily reached on foot, where you occasionally encountered secluded temples along the paths.
Zaragoza in Spain. Walked from the old city centre to the edge of the city (which was apartment blocks) in a reasonable time. It was mostly agricultural land, but I recall some forest along the river.
San Sebastián, also in Spain. You can walk into the hills, but we went to the beach.
Parks covered in homeless people is a US problem, in my experience. I've been around most of Europe and haven't seen anything like that. The homeless problem in the US is a policy choice, just like the urban development is. It's easily solvable given political will.
Can you walk to a cafe?